Class 15: OutlineLast Time:Magnetic Fields& Magnetic DipolesMagnetic FieldsWhat Kind of Motion Does this Lead to?Cyclotron MotionCurrent Carrying WiresMagnetic Force on Current-Carrying WireDemonstration:Jumping WireMagnetic Force on Current-Carrying WirePRS Questions:5 Predictions For Experiment 6Experiment 6:Magnetic ForceMid-term Course EvaluationLab Summary:Currents FEEL Forces in Magnetic FieldsQuestion:What happens if currents are next to each other?Demonstration:Parallel & Anti-Parallel CurrentsHow Do They Interact?Sources of Magnetic Fields:Biot-SavartElectric Field Of Point ChargeMagnetic Field Of Moving ChargeThe Biot-Savart LawThe Right-Hand Rule #2Animation: Field Generated by a Moving Charge (http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/01-MovingCDemonstration:Field Generated by WireExample : Coil of Radius RExample : Coil of Radius RExample : Coil of Radius RExample : Coil of Radius RExample : Coil of Radius RPRS Questions:B fields Generated by CurrentsGroup Problem:B Field from Coil of Radius RField Pressures and Tensions:A Way To Understand the qVxB Magnetic ForceTension and Pressures Transmitted by E and BExample of E Pressure/TensionExample of B Pressure/Tension1P15-Class 15: OutlineHour 1:Magnetic ForceExpt. 6: Magnetic ForceHour 2:Creating B Fields: Biot-Savart2P15-Last Time:Magnetic Fields& Magnetic Dipoles3P15-Magnetic FieldsBq=×FvBGGGMagnetic Dipoles Create and Feel B Fields:Also saw thatmoving charges feel a force:4P15-What Kind of Motion Does this Lead to?5P15-Cyclotron Motion(1) r : radius of the circle 2mvqvBr=mvrqB⇒=(2) T : period of the motion 22rmTvqBππ==2vqBfrmωπ===(3) ω : cyclotron frequency6P15-Current Carrying Wires7P15-Bq=×FvBGGGMagnetic Force on Current-Carrying Wire()mcharges=×BG()BI=×FLBGGGchargems=×BG8P15-Demonstration:Jumping Wire9P15-Magnetic Force on Current-Carrying WireCurrent is moving charges, and we know that moving charges feel a force in a magnetic field10P15-PRS Questions:5 Predictions For Experiment 611P15-Experiment 6:Magnetic Force12P15-Mid-term Course Evaluation13P15-Lab Summary:Currents FEEL Forces in Magnetic FieldsQuestion:What happens if currents are next to each other?14P15-Demonstration:Parallel & Anti-Parallel Currents15P15-How Do They Interact?Moving charges also create magnetic fields!The current in one wire creates a magnetic field that is felt by the other wire. This is the rest of today’s focus(http://ocw.mit.edu/ans7870/8/8.02T/f04/vis(http://ocw.mit.edu/ans7870/8/8.02T/f04ualizations/magnetostatics/13-/visualizations/magnetostatics/14-ParallelWires/13-SeriesWires/14-Series_320.htmlParallel_Wires_320_f185.html))16P15-Sources of Magnetic Fields:Biot-Savart17P15-Electric Field Of Point ChargeAn electric charge produces an electric field: rˆ21ˆ4oqrπε=ErGrˆ: unit vector directed from q to P18P15-Magnetic Field Of Moving ChargeMoving charge with velocity v produces magnetic field:2ˆx4oqrµπ=vrBGG:ˆrrˆPunit vector directed from q to Ppermeability of free space70410Tm/Aµπ−=× ⋅19P15-The Biot-Savart LawCurrent element of length ds carrying current I produces a magnetic field:20ˆ4 rdI rsBd×=GGπµ(http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.html)20P15-The Right-Hand Rule #2ˆˆˆ×=z ρφ21P15-Animation: Field Generated by a Moving Charge (http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/01-MovingChargePosMag/01-MovChrgMagPos_f223_320.html)22P15-Demonstration:Field Generated by Wire23P15-Example : Coil of Radius RConsider a coil with radius R and current IIIIPFind the magnetic field B at the center (P)24P15-Example : Coil of Radius RConsider a coil with radius R and current IIIIP1) Think about it:• Legs contribute nothingI parallel to r• Ring makes field into page2) Choose a ds3) Pick your coordinates4) Write Biot-Savart25P15-Example : Coil of Radius RIn the circular part of the coil…ˆd⊥srGˆ|d | ds→×=srGrˆsGdIII02ˆ4dIdBrµπ×=srGBiot-Savart:024Idsrµπ=024IRdRµθπ=04IdRµθπ=26P15-Example : Coil of Radius RConsider a coil with radius R and current I04IddBRµθπ=2004IdBdBRπµθπ==∫∫()2000244IIdRRπµµθπππ==∫IIIθsGd0into page2IRµ=BG27P15-Example : Coil of Radius RNotes:•This is an EASY Biot-Savart problem:• No vectors involved•This is what I would expect on examIIPIpage into20RIµ=BG28P15-PRS Questions:B fields Generated by Currents29P15-Group Problem:B Field from Coil of Radius RConsider a coil with radius R and carrying a current IWARNING:This is much harder than what I just did! Why??What is B at point P?30P15-Field Pressures and Tensions:A Way To Understand the qVxB Magnetic Force31P15-Tension and Pressures Transmitted by E and BFields (E or B):• Transmit tension along field direction (Field lines want to pull straight)• Exert pressure perpendicular to field (Field lines repel)32P15-Example of E Pressure/Tension(http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/11-forceq/11-ForceQ_f0_320.html)Positive charge in uniform (downward) E fieldElectric force on the charge is combination of 1. Pressure pushing down from top 2. Tension pulling down towards bottom33P15-Example of B Pressure/TensionPositive charge moving out of page in uniform (downwards) B field. Magnetic force combines:1. Pressure pushing from left 2. Tension pulling to
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